Let the function $f:[0,2] \rightarrow \mathbb{R}$ be defined as $f(x)= \begin{cases}e^{\min \left\{x^{2}, x-[x]\right\},} & x \in[0,1) \\ e^{\left[x-\log _{e} x\right]}, & x \in[1,2]\end{cases}$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int_\limits{0}^{2} x f(x) d x$ is :

Question

Let the function $f:[0,2] \rightarrow \mathbb{R}$ be defined as

$f(x)= \begin{cases}e^{\min \left\{x^{2}, x-[x]\right\},} & x \in[0,1) \\ e^{\left[x-\log _{e} x\right]}, & x \in[1,2]\end{cases}$

where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\int_\limits{0}^{2} x f(x) d x$ is :

$2 e-1$

$2 e-\frac{1}{2}$

$1+\frac{3 e}{2}$

$(e-1)\left(e^{2}+\frac{1}{2}\right)$

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